Gravitational Fields

Chapter 6

Analytic calculation of gravitational ﬁelds is easy for spherical systems, but few general results are available for non-spherical mass distributions. Mass models with tractable potentials may however be combined to approximate the potentials of real galaxies.

6.1 Conservative Force Fields

In a one-dimensional system it is always possible to deﬁne a potential energy corresponding to any µ given f ´x ;let

Z x

¼ ¼

µ= ´ µ ; U´x dx f x (6.1) x0

where x0 is an arbitrary position at which U = 0. Different choices of x0 produce potential energies differing by an additive constant; this constant has no inﬂuence on the dynamics of the system.

In a space of n > 1 dimensions the analogous path integral,

Z x ¼

n¼

¡ µ; µ= ´ U ´x dxfx(6.2) x0 may depend on the exact route taken from points x0 to x; if it does, a unique potential energy cannot

be deﬁned. One condition for this integral to be path-independent is that the integral of the force

µ ´ µ f ´x around all closed paths vanishes. An equivalent condition is that there is some function U x

such that

µ= ∇: f´x U(6.3)

Force fields obeying these conditions are conservative. The gravitational field of a stationary point mass is an example of a conservative field; the energy released in moving from radius r1 to radius

r2 < r1 is exactly equal to that consumed in moving back from r2 to r1.

6.2 General Results

In astrophysical applications it’s natural to work with the path integral of the acceleration rather µ than the force; this integral is the potential energy per unit mass or gravitational potential, Φ´x ,

46 CHAPTER 6. GRAVITATIONAL FIELDS

Φ´ µ ρ ´ µ and the potential energy of a test mass m is just U = m x . For an arbitrary mass density x ,the

potential is

¼ µ ρ´x

3¼

µ=

Φ´x Gdx(6.4)

j jxx where G is the gravitational constant and the integral is taken over all space. Poisson’s equation provides another way to express the relationship between density and potential:

2 π ρ : ∇ Φ = 4 G (6.5) ρ Note that these relationships are linear;ifρ1 generates Φ1 and ρ2 generates Φ2 then ρ1 · 2 Φ generates Φ1 · 2. Gauss’s theorem relates the mass within some volume V to the gradient of the ﬁeld on its surface

dV:

Z Z

3 2

µ= ¡ ∇Φ ; 4πG d xρ ´x d S (6.6) V dV where d2S is an element of surface area with an outward-pointing normal vector.

6.3 Spherical Potentials

Consider a spherical shell of mass m; Newton’s ﬁrst and second theorems (BT87, Ch. 2.1) imply

¯ the acceleration inside the shell vanishes, and

= ¯ the acceleration outside the shell is Gm r .

From this, it follows that the potential of an arbitrary spherical mass distribution is Z

rZ r µ

Φ M´x µ= ´ µ= ; ´r dxa x Gdx 2 (6.7) r0r0x where the enclosed mass is Z r

µ= πρ ´ µ : M ´r 4dxx x (6.8)

∞ = ∞ Φ´ µ < If M converges for r ! , it’s convenient to set r0 ;then r 0 for all ﬁnite r.

6.3.1 Elementary Examples

A point of mass M has the potential

Φ M µ= : ´r G(6.9) r

This is known as a Keplerian potential since orbits in this potential obey Kepler’s three laws. A

µ= = circular orbit at radius r has velocity vc ´r GM r.

A uniform sphere of mass M and radius a has the potential ´

πρ´ = µ; <

2Gar3ra µ=

Φ´r (6.10)

= ; > GM r r a

=´ π = µ where ρ = M 4 a 3 is the mass density. Outside the sphere the potential is Keplerian, while inside it has the form of a parabola; both the potential and its derivative are continuous at the surface of the sphere. 6.4. AXISYMMETRIC POTENTIALS 47

µ=ρ´ = µ A singular isothermal sphere with density proﬁle ρ ´r 0rr0has the potential

µ= πρ´ = µ : Φ´

r 4G0r0ln r r0 (6.11) Õ π ρ 2 The circular velocity vc = 4 G 0r0 is constant with radius. This potential is often used to approximate the potentials of galaxies with ﬂat rotation curves, but some outer cut-off must be imposed to obtain a ﬁnite total mass.

6.3.2 Potential-Density Pairs

Pairs of functions related by Poisson’s equation provide convenient building-blocks for galaxy models. Three such functions often used in the literature are listed here; all describe models characterized

by a total mass M and a length scale a:

µ Φ´ µ

Name ρ ´r r

2 5=2

3M r GM Ô Plummer 1 · π 3 2 2 2 4 a a r · a

M a GM

Hernquist 3

· µ ·

2π r ´r a r a M a GM a

Jaffe 2 2 ln

· µ ·

4π r ´r a a r a

h i

¡ γ

4 1 r 2

γ 6= ;

γ µ

´3 M a GM 1 2 ·

γ 2 r a ¡ Gamma γ

3 γ 4

· µ

4πa r ´r a a r γ = ln ; 2 r ·a

The Plummer (1911) density proﬁle has a ﬁnite-density core and falls off as r 5 at large radii; this is a steeper fall-off than is generally seen in galaxies. Hernquist (1990) and Jaffe (1983) models, on the other hand, both decline like r 4 at large radii; this power law has a sound theoretical basis in the mechanics of violent relaxation. The Hernquist model has a gentle power-law cusp at small radii,

while the Jaffe model has a steeper cusp. Gamma models (Dehnen 1993; Tremaine et al. 1994) γ =

include both Hernquist (γ = 1) and Jaffe ( 2) models as special cases; the best approximation to = the de Vaucouleurs proﬁle has γ = 3 2.

6.4 Axisymmetric Potentials

If the mass distribution is a function of two variables, cylindrical radius R and height z, the problem of calculating the potential becomes a good deal harder. A general expression exists for inﬁnitely thin disks, but only special cases are known for systems with ﬁnite thickness.

6.4.1 Thin disks µ An axisymmetric disk is described by a surface mass density Σ´R . Potential-surface density expres-

sions for several important cases are collected here:

µ Φ´ ; µ Name Σ´R R z

aM GM

Kuzmin Ô

2 2 3=2 2 2

π ´ · µ

·j jµ

2 R a R ·´az

d n 1 d n 1 Toomre Σ Φ da2 Kuzmin da2 Kuzmin

µ ´ j jµ ´ µ Bessel J ´kR exp k z J kR 2πG 0 0 48 CHAPTER 6. GRAVITATIONAL FIELDS

Figure 6.1: Surgery on the potential of a point mass produces the potential of a Kuzmin disk. Left:

potential of a point mass. Contours show equal steps of Φ∝1=r, while the arrows in the upper left ¦

quadrant show the radial force field. Dotted lines show z = a. Right: potential of a Kuzmin disk, j produced by excising the region jz afrom the field shown on the left. Arrows again indicate the force field; note that these no longer converge on the origin.

The potential of a Kuzmin (1956) disk may be constructed by performing surgery on the ﬁeld

of a point of mass M (6.9), as shown in Fig. 6.1. Let the z coordinate be the axis of the disk, and

> < join the field for z > a 0 directly to the field for z a. The result satisfies Laplace’s equation,

6= ∇ Φ = 0, everywhere z 0; thus the matter density which generates this potential is conﬁned to the

disk plane z = 0. Gauss’s theorem (6.6) can be used to ﬁnd the required surface density. Consider an inﬁnitesimal box which straddles the disk plane at radius R, and let the top and bottom surfaces

2 2 3 =2

´ · µ

of this box have area δA. The integral of ∇Φ over the surface of the box is 2GMaδA = R a ,

µ Σ and mass within the box is δAΣ´R . Setting these two integrals equal yields the surface density K of Kuzmin’s disk.

Toomre (1962) devised an inﬁnite sequence of models, parameterized by the integer n.The = n= 1 Toomre model is identical to Kuzmin’s disk, while the n 2 model is found by differentiating the potential and surface density of Kuzmin’s disk by the parameter a2; the linearity of Poisson’s ∞ equation (6.5) guarantees that the result is a valid potential-surface density pair. In the limit n ! the surface density is a gaussian in R. The ‘Bessel’ potential-density pair, also due to Toomre (1962), describes another ﬁeld which obeys Laplace’s equation everywhere except on the disk plane. Unlike other cases, the surface

density – once again obtained using Gauss’s theorem – is not positive-deﬁnite. What good is it? A µ

disk with arbitrary surface density Σ´R may be expanded as Z

Σ µ= ´ µ ´ µ

´R dkS k kJ0 kR (6.12)

µ Σ´ µ

where S ´k is the Hankel transform of R ,

µ= Σ´ µ ´ µ : S ´k dR R RJ0 kR (6.13) 6.4. AXISYMMETRIC POTENTIALS 49

The potential generated by this disk is then Z

Φ ; µ= π´ j jµ ´ µ ´ µ : ´R z 2Gdk exp k z J0 kR S k (6.14)

This method may be used to evaluate the potential of an exponential disk. Consider a disk with surface density

α2M

α α

µ= =Σ; Σ´R ee(6.15) 2π0 where M is the total disk mass and α is the inverse of the disk scale length. The resulting potential

j j

∞ k z µ

2πGΣ0J0´kR e

; µ= : Φ´R z dk (6.16)

22 2

· α ] α0[1 k

In the plane of the disk, z = 0 and this integral becomes

; µ= πΣ[ ´ µ ´ µ ´ µ ´ µ] ;

Φ´R 0 G0RI0yK1yI1yK0y(6.17) α = where y = R 2, and In and Kn are modiﬁed Bessel functions of the ﬁrst and second kind.

6.4.2 Flattened Systems

Potential-density pairs for a few ﬂattened systems are listed here:

; µ Φ´ ; µ

Name ρ ´R z R z

Ô Ô

2 2 2 2 2 2

·´ · · µ´ · · µ

b M aR a 3 z b a z b GM

Õ Ô

Miyamoto-Nagai π Ô =

4 2 2 2 2 5=2 2 2 3 2

·´ · · µ ] ´ · µ

[R a z b z b 2 2 2 2

· · µ

R ·´a b z

n n d Φ d Φ

Satoh db2 MN db2 MN

2 2 2 2 2 2

· µ · ·´ µ

v0 ´2q 1 Rc R 2 q z 1 2 2 2 2 2

´ · · = µ

Logarithmic 2 2 2 2 2 2 v ln Rc R z q

π ´ · · µ 4 Gq Rc R z q 2 0

The Miyamoto & Nagai (1975) potential includes the Plummer model (if a = 0) and the Kuzmin

disk (if b = 0); thus it can describe a wide range of shapes, from a spherical system to an inﬁnitely thin disk. Satoh’s (1980) models are derived from the Miyamoto-Nagai model by differentiating with respect to the parameter b2, much as Toomre’s models are derived from Kuzmin’s. The logarithmic potential is often used to describe galaxies with approximately ﬂat rotation

Ô 2 2

= = · curves; in the z = 0 plane, the circular velocity vc v0R Rc R rises linearly for R Rc and levels off for large R. The corresponding density can be found by inserting the potential in Poisson’s

2 = equation. As Fig. 2.8 of BT87 shows, this density is ‘dimpled’ at the poles; if q < 1 2 the density

must actually be negative along the z axis, which is unphysical. This dimpling is demanded by the

; µ ! ∞ assumption that Φ´R z is stratiﬁed on surfaces of constant ellipticity as R . In models where the potential becomes more spherical at larger radii this dimpling isn’t necessary. A great deal of analytic machinery exists to calculate potentials for systems with densities strati- ﬁed on ellipsoidal surfaces (BT87, Ch. 2.3). Much of this machinery generalizes to triaxial systems; a few key results are given below. 50 CHAPTER 6. GRAVITATIONAL FIELDS

6.5 Triaxial Potentials

6.5.1 Ellipsoidal systems

The triaxial generalization of an inﬁnitely thin spherical shell is an inﬁnitely thin homoeoid,which 2 2 2 has constant density between surfaces m and m · dm ,where

2 2

2 2 y z

· · : m = x (6.18) b2 c2

Φ : Just as in the spherical case, the acceleration inside the shell vanishes; thus = const inside the shell and on its surface. Outside the shell, the potential is stratiﬁed on ellipsoidal surfaces deﬁned by 2 2 2

2 x y z

· ·

m = 2 2 (6.19)

τ · τ · τ

1 · b c τ = where the parameter τ > 0 labels the surface. Notice that for 0 the isopotential surface coin- ∞ cides with the surface of the homoeoid, while in the limit τ ! the isopotential surfaces become spherical.

ρ ´ µ

A triaxial mass distribution with ρ = m is equivalent to the superposition of a series of thin

; ; µ homoeoids. The acceleration at a given point ´x0 y0 zo is generated entirely by the mass within

2 2 2 2 2 2 2

= · = · = m < m0 x0 y0 b z0 c , just as in a spherical system the acceleration at r is due only to the mass within r. Since the isopotential surfaces of a homoeoid become spherical at large r,mass 2 lying within m m0 generates a nearly radial force ﬁeld, while mass lying only slightly within 2 m0 generates a more non-radial force ﬁeld. Thus the potential of a centrally concentrated body is always more spherical than the potential of uniform body with the same axial ratios.

6.5.2 Logarithmic potential

The triaxial version of the logarithmic potential is 2 2

Φ 122 2 y z ; ; µ= · · · : ´x y z vln R x (6.20) 20c b2 c2

As in the 2-D case above, the corresponding density is not guaranteed to be positive if the potential is too strongly ﬂattened. Nonetheless, the logarithmic potential is a useful approximation when studying orbits in triaxial galaxies.

Problems

6.1. Evil space aliens compress the Earth to an inﬁnitely thin disk of uniform surface density ¨ without changing its radius. What is the gravitational acceleration experienced by a survivor stand-

ing on the surface of this disk? µ 6.2. Figure 2-17 of BT87 compares circular-speed curves vc ´r for three mass distributions: “an

exponential disk (full curve); a point with the same total mass (dotted curve); the spherical body for

µ ´ µ which M ´r is [identical to M r for the exponential disk] (dashed curve)” (see BT87, p. 78). In qualitative terms, ﬁrst explain why the dashed curve lies below the dotted curve at all radii. Then, explain why the full curve lies above the dotted curve at some radii.